Modular resurgence, $q$-Pochhammer symbols, and quantum operators from mirror curves

TL;DR

This paper explores the resurgence and modularity of $q$-Pochhammer symbols, constructing new resurgent series and connecting them to quantum operators in topological string theory, providing evidence for existing conjectures.

Contribution

It introduces a new family of modular resurgent series from $q$-Pochhammer sums and links them to quantum operators in topological string/spectral theory.

Findings

Constructed infinite families of modular resurgent series.

Linked spectral traces of quantum operators to $q$-Pochhammer sums.

Demonstrated strong-weak resurgent symmetry in various local $ ext{P}^{m,n}$}.

Abstract

Building on the results of [1,2], we study the resurgence of $q$-Pochhammer symbols and determine their summability and quantum modularity properties. We construct a new, infinite family of pairs of modular resurgent series from the asymptotic expansions of sums of $q$-Pochhammer symbols weighted by suitable Dirichlet characters. These weighted sums fit into the modular resurgence paradigm and provide further evidence supporting our conjectures in [1]. In the context of the topological string/spectral theory correspondence for toric Calabi-Yau threefolds, Kashaev and Mari\~no proved that the spectral traces of canonical quantum operators associated with local weighted projective planes can be expressed as sums of $q$-Pochhammer symbols. Exploiting this relation, we show that an exact strong-weak resurgent symmetry, first observed by the second author in [3] and fully formalized in [2] for local $\mathbb{P}^2$, applies to all local $\mathbb{P}^{m,n}$, albeit stripped of some of the underlying number-theoretic properties. Under some assumptions, these properties are restored when considering linear combinations of the spectral traces that reproduce the weighted sums above.